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# Diffraction Gratings

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Diffraction Gratings

Suppose we had a light bulb that emitted exactly four frequencies of light; one frequency in each of the colors red, yellow, green, and blue.  To our eye, this bulb would appear white because the combination of those four colors produces white light. If viewed through a diffraction grating, however, each color of light would be visible. The original white light bulb is visible in the center of the image, and interference causes the light bulb to appear in each color to the left and the right.

### Diffraction Gratings

Diffraction gratings are composed of a multitude of slits lined up side by side, not unlike a series of double slits next to each other. They can be made by scratching very fine lines with a diamond point on glass, or by pressing plastic film on glass gratings so that the scratches are replicated. The clear places between the scratches behave as slits similar to the slits in a double slit experiment and the gratings form interference patterns in the same general way that double slits do. With more slits, there are more light waves out of phase with each other, causing more destructive interference. Compared to the interference pattern of a double slit, the diffraction grating interference pattern's colors are spread out further and the dark regions are broader. This allows for more precise wavelength determination than with double slits. The image below shows the diffraction pattern emanating from a white light.

Also in this image is the measurement for θ, which can be used to calculate the wavelength of the original light source. The equation from the double slit experiment can be adjusted slightly to work with diffraction gratings. Where λ is the wavelength of light, d  is the distance between the slits on the grating, and θ is the angle between the incident (original) light and the refracted light,

$\lambda=\frac{xd}{L}=d \sin \theta$  (Note that $\frac{x}{L} = \sin \theta$ , using the small angle approximation theorem.)

Looking at the equation, $x=\frac{\lambda L}{d}$ , it should be apparent that as the distance between the lines on the grating become smaller and smaller, the distance between the images on the screen will become larger and larger.  Diffraction gratings are often identified by the number of lines per centimeter; gratings with more lines per centimeter are usually more useful because the greater the number of lines, the smaller the distance between the lines, and the greater the separation of images on the screen.

Example Problem: A good diffraction grating has 2500 lines/cm. What is the distance between two lines on the grating?

Solution:   d   $=\frac{1}{2500 \ cm^{-1}}=0.00040 \ cm$

Example Problem: Using a diffraction grating with a spacing of 0.00040 cm, a red line appears 16.5 cm from the central line on the screen. The screen is 1.00 m from the grating. What is the wavelength of the light?

Solution:   $\lambda=\frac{xd}{L}=\frac{(0.165 \ m)(4.0 \times 10^{-6} \ m)}{1.00 \ m}=6.6 \times 10^{-7} \ m$

#### Summary

• Diffraction gratings can be made by blocking light from traveling through a translucent medium; the clear places behave as slits similar to the slits in a double slit experiment.
• Diffraction gratings form interference patterns much like double slits, though brighter and with more space between the lines.
• The equation used with double slit experiments to measure wavelength is adjusted slightly to work with diffraction gratings.  $\lambda=\frac{xd}{L}=d \sin \theta$

#### Practice

1. How does a diffraction grating differ from single or double slit?
2. What happens when you increase the number of slits in a diffraction grating?

#### Review

1. White light is directed toward a diffraction grating and that light passes through the grating, causing its monochromatic bands appear on the screen. Which color will be closest to the central white?
2. Three discrete spectral lines occur at angles of 10.1°, 13.7°, and 14.8° respectively in the first order spectrum. If the grating has 3660 lines/cm, what are the wavelengths of these three colors of light?
3. A 20.0 mm section of diffraction grating has 6000 lines. At what angle will the maximum bright band appear if the wavelength is 589 nm?
4. Laser light is passed through a diffraction grating with 7000 lines/cm. The first order maximum on the screen is 25° away from the central maximum. What is the wavelength of the light?